Dynamic Competitive Revenue Management with

Forward and Spot Markets

Srinivas Krishnamoorthy

Guillermo Gallego

Columbia University

Robert Phillips

Nomis Solutions

Entrant

1010 pppp

1p 1p

Entrant

0Capacity C

0p 0p

101 ppp

0p

Entrant

0Capacity C

1p 1p

0p

0Capacity C

10 pp

1p

Entrant

1p

Motivation

Incumbent

Demand

D

Buyer

E[D] =

1Capacity C

Example Applications

Buyer

• OEM • Utility company• Tour operator• Freight consolidator• Ad agency

Capacity providers

• Contract manufacturers• Power plants• Airlines• Freight carriers• TV Networks

Related Literature

Competitive Revenue Management & Pricing

• Perakis & Sood (2002, 2003)

• Netessine & Shumsky (2001)

• Li & Oum(1998)• Talluri (2003)

CompetitiveNewsvendor

• Parlar (1988)• Karjalainen (1992) • Lippman & McCardle

(1997)• Mahajan & van Ryzin

(1999)• Rudi & Netessine (2000) • Dana & Petruzzi (2001)

Model

• A buyer faces random demand D• Two providers with capacities C0 and C1

• Entrant offers forward price p0 and spot price p0

• Incumbent offers forward price p1 and spot price p1 • Prices satisfy p0 < p1 < p0 < p1

• Entrant’s decision - offer C0 units forward • Incumbent’s decision - offer C1 units forward• Buyer’s decision - buy forward x units from entrant and y

units from incumbent• Buyer satisfies any excess demand by buying in spot

market

The Buyer’s Problem

• Buyer’s cost = entrant’s revenue + incumbent’s revenue

• Buyer minimizes expected cost

Optimal solution (x*, y*) depends on C0, C1

),(ˆ),(ˆ),( 10 yxyxyxc

],)[(Min),(ˆ 0000 xCyxDEpxpyx

],)[(Min),(ˆ 10111 yCyCDEpypyx

ZyxCyCx

yxc

,,0,0 s.t.

),(min

10

The Providers’ Problems

• Entrant maximizes expected revenue

• Incumbent maximizes expected revenue

)),(),,((ˆ),(where

,0 s.t.

),(max

10*

10*

0100

000

100

CCyCCxCC

ZCCC

CC

)),(),,((ˆ),(where

,0 s.t.

),(max

10*

10*

1101

111

101

CCyCCxCC

ZCCC

CC

Game Between Providers

• Forward and spot prices are fixed.• Entrant and incumbent simultaneously announce

forward capacities C0 and C1 respectively. – Entrant attempts to maximize 0(C0,C1).

– Incumbent attempts to maximize 1(C0,C1).

• Buyer determines forward purchases x*, y* that minimize c(x,y).

• After forward purchasing, she observes total demand D and satisfies any excess demand in the spot market.

Buyer’s Market

C0

C1

= 50

C0 = 50

C1 = 100

(0,10)

(0,41)

(0,0)

(43,11)

(50,0)

Market in Flux

C0

C1

= 100

C0 = 50

C1 = 100

(0,59)

(0,87)

(0,0)

(46,60)

(24,0)

(23,64)

(46,87)

(46,0)

Providers’ Market

C0

C1

= 150

C0 = 50

C1 = 100

(0,0)

The Repeated Game

• The game is now played repeatedly an infinite number of times (e.g. two airlines may compete for passengers daily on a particular route)

• Each provider’s revenue is the present value of the revenue stream from the infinite sequence of stage games

• Can each provider obtain higher revenue then under the single stage Nash equilibrium?

• If so, then what is the strategy to be followed by the providers?

The Different Market Regimes

C0

C1

(0,10)

(0,41)

(0,0)

(43,11)

(50,0)C0

C1

(0,59)

(0,87)

(0,0)

(46,60)

(24,0)

(23,64)

(46,87)

(46,0)

C0

C1

(0,0)

Buyer’s Market ( = 50) Market in Flux ( = 100)

Providers’ Market ( = 150)

C0 = 50

C1 = 100

Feasible Revenues

Feasible revenues are convex combinations of pure strategy revenues.

Lemma

There exists a feasible revenue that yields revenues (z0, z1) with z0 > f0 and z1 > f1

)0,(strategy pure with therevenues ),(

),0(strategy pure with therevenues ),(

)0,0(strategy pure with therevenues ),(

),(strategy pure with therevenues ),(

0

1

10

10

10

10

10

f

f

ff

Cee

Cii

ss

CCff

Subgame – Perfect Nash Equilibrium

Theorem

For discount rates sufficiently close to 1 there exists a subgame-perfect Nash equilibrium for the infinite game that achieves average revenues

(z0, z1) with z0 > f0 and z1 > f1

Proof

From Lemma (previous slide) and Friedman’s Theorem (1971) for repeated games

Trigger Strategy

If (Cz0, Cz1

) is the collection of actions that yields (z0, z1) as the average revenues per stage, then the subgame – perfect equilibrium can be achieved by the following strategy for the entrant (incumbent) :

Play Cz0 (Cz1) in the first stage. In the tth stage, if the

outcome of all the preceding stages has been (Cz0, Cz1

),

then play Cz0 (Cz1), otherwise play Cf0 (Cf1

).

),)(1(),(),( 101010 iisszz

Obtaining Higher Revenues in a Buyer’s Market

(e0, e1) (8100, 2165)2000

2200

2400

2600

2800

3000

3200

3400

3600

7500 8000 8500 9000 9500 10000

Entrant

Incumbent

(z0, z1) (8920, 2837)

(s0, s1) (9917, 2165)

(f0, f1) (7949, 2176)

(i0, i1) (7923, 3508)

100

50

59

1

0

C

C

(e0, e1) = 8100, 2165)

Numerical Results (Buyer’s Market)

(Cf0, Cf1

) (f0, f1) (z0, z1) *

30 (23, 5) (5124, 0.15) (5500, 450) 0.50 0.891

45 (38, 8) (7390, 212) (8100, 836) 0.50 0.903

50 (43, 11) (7828, 648) (8598, 1339) 0.50 0.897

54 (47, 14) (7954, 1248) (8836, 1910) 0.50 0.903

59 (46, 19) (7949, 2176) (8920, 2837) 0.50 0.901

Market in Flux

The two providers obtain revenues (m0, m1) at a mixed strategy equilibrium (0, 1 ) for the stage game

Proposition

There exists a convex combination of the revenues (s0, s1) and (i0, i1) that yields revenues (z0, z1) with z0 > m0 and z1 > m1

))(),0(())(),0((10 111000 ff CC

Subgame – Perfect Nash Equilibrium(Market in Flux)

TheoremFor discount factors sufficiently close to 1 there exists a subgame perfect Nash equilibrium for the infinite game that achieves average revenues

(z0, z1) with z0 > m0 and z1 > m1

(The subgame perfect equilibrium can once again be achieved by a trigger strategy similar to the strategy for a Buyer’s Market.)

Numerical Results (Market in Flux)

(0,Cf0) 0(0) (0,Cf1

) 1(0) (m0, m1) (z0, z1) *

60 (0,46) 0.002 (0, 50) 0.763 (8074, 2373) (8362, 3699) 0.80

75 (0.47) 0.099 (0, 64) 0.751 (8080, 5751) (8455, 6905) 0.65

90 (0,46) 0.153 (0, 78) 0.748 (8106, 9200) (8497, 10168) 0.63

100 (0,46) 0.299 (0, 87) 0.741 (8102, 11500) (8536, 12332) 0.56

115 (0,46) 0.450 (0, 100) 0.726 (8103, 14949) (8614, 15559) 0.44

( = 0.80)

Concluding Remarks

• We have analyzed a revenue management game with two providers selling in a forward and a spot market to a single buyer making bulk purchases

• Competitive considerations can motivate capacity providers to sell in a discounted forward market even when buyers’ willingness-to-pay is the same in both the forward and the spot market

• For the static game there are three market regimes: Buyer’s Market (Low Demand)Market in Flux (Moderate Demand)Providers’ Market (High Demand)

• The two providers can increase their average revenues above their static Nash equilibrium revenues by implicit collusion when the game is played repeatedly